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1963 IEEE T R A N S A C T I O N S ON E L E C T R O N D E V I C E
S
Concentration-Dependent Diffusionof Boronand Phosphorus in
Silicon*
J. J . CHANGt, MEMBER, IEEE
Summary-A calculation is made of the tail of the impurity
con-centration profile resulting from concentration-dependent
dif-fusion from a constant surface concentration into a
semi-infinitemedium. The calculation predicts that if the
concentration de-pendence at low impurity concentrations is
negligible, the low con-centration portion of the dopingprofile
should still take he fa-miliar form, C = C,erfc ( x / Z DW). D Zs
the commonly knowndiffusion coefficient at low impurity
concentrations, while C, is theapparent surface concentration. C,
depends on he actual surfaceconcentration and also depends on how
the diffusioncoefficientvaries with impurity concentration at high
concentrations. It is aconstant for a given diffusion system but
could be orders of magni-tude higher than the actual surface
concentration. Empirical datahave been obtained for boron and
phosphorus diffusions in siliconand found to be in good agreement
with this prediction.
INTRODUCTIONISCREPANCIES between theory and experiment
in high temperaturend high concentrationdiffusion of impurities
in semiconductors have
attracted the attention of many authors.1-6 In the caseof
phosphorus diffusion in silicon, Tannenbaum foundthat 1) the
diffusioncoefficient is aconstantup t o aconcentration of about 10
cm-3 but is a strong functionof concentration above th at value,
and 2) the concentra-tion profile does not obey the complementary
error func-tion predicted by simple theory. At the present time,
nogeneral theory is available for the calculation of
theconcentration dependence. Thus, a theoretical calculationof the
entire concentration profile is impossible. Thisdifficulty hinders
the work of those engaged in designingand analyzing semiconductor
devicesormed by highconcentration diffusions.It is the purpose of
this paper t o point out th at theconcentration-dependent formula
given by Crank reducesto a very simple form when applied to the
typeof systemobserved by Tannenbaum where the concentration
1963.* Received July 2, 1963; revised manuscript received July
24,t Bell Telephone Laboratories, Inc., Murray Hill, N. J.J. W.
Allen and F. A. Cunnell, Diffusion of zinc in galliumarsenide,
Nature (Correspondence), vol. 182,p. 1158; October, 1958.* E.
Tannenbaum, q,etailed analysis of thin phosphorus-diffusedlayers in
p-type silicon, Solid-StateElectron., vol. 2, pp. 123-132;March,
1961.
Double-Diffusedilicon Structures, presented at Solid-statea R.
C. Musa, Concentration Dependent Diffusion Rates inDevice Research
Conference, Stanford, Calif . June 26-28, 1961.* W.
Shockley,F$ld-enhanced donor d h s i o n in degeneratesemiconductor
layers, J. Appl. Phys. (Correspondence), vol.32,t>ributionof
phosphorus atoms during diffusion in silicon, SovietV. R.
Subashiev, A. P. Landsman, and A. A. Kukharskii, Dis-Phys.
Solid-State, vol. 2, pp. 2406-2411; May, 1961.
6 J. Crank, TheMathematics of Diffusion, The CIarendonPress,
Oxford, England, pp. 148-150; 1956.
pp. 1402-1403; July, 1961.
357
dependence of the diffusion oefficientsnegligible atlow impurity
concentrations.CALCULATION
Crankshowed th at when the diffusioncoefficient Ddepends only on
the concentration C of the diffusant, thefollowing formula applies
t o the diffusion from a constantsurface concentration into a
semi-infinite medium:_ - iy1/@ exP [-lV2y/6) d y ] dy- 1 -C, .
(1)La 1/6) exp[-c2 y f / 6 ) d y ] dyIn ( l ) ,C, is the actual
surface concentration, 6 is theratio of the diffusion coefficientD
to the ow concentrationdiffusion coefficient D, nd y is defined
as
y = ~ / 2 D { / t ~ / ~ , (2)where z is the distance from the
surface and t is the dif-fusion time.
We consider the case in which 6 = 1 for concentrationsbelow a
certain value C, and is concentration-dependentabove this value.
(In the case of phosphorus diffusion insilicon, ccording to
Tannenbaum, C, N _ 10 CM-~).When (1) is evaluated numerically, one
obtains C as afunction of y. If C = C, when y = y. and C < C,
wheny > yc, then for y > yc, (1) can be written as
?l= Cr n exp [-/I 2y/6) dy - I:y d y ] dy * Y
210- g ~ ~xp [y: - 1 2 y f / 6 ) d y ] erfc y, (8)and
similarly
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358 IEEE TRANSACTIONS ON ELECTRON DEVICES Novemben55 LOG00ULL0
LOGrka4s
TI
STRONGCONCENTRATIONDEPENDENCECONCENTRATIONNEGLIGIBLEDEPENDENCE
-- - - _ _ _ _ _ _ _ _
DISTANCE FROM SURFACEFig. 1-Impurity profile of
concentration-dependent diffusion.
Substitution of (61, (8) and (9) into (3 ) yieldsC(y) = C: erfc
g , (C < C ,), (10)where C: is the apparent surface
concentration given by
% = erfc y. + 2T-12C, exp [LUC2Y/6) dY - y:]-i1/6) exp [ s, ( 2
g / 6 ) d y ] dy. (11)If 6 = 1 for all concentrations
(concentration indepen-dent case), the right-hand side of (11)is
unity, thenC: =C, and (10) reduces t o the well-known
concentrationindependent formula. Since 6 2 1 n the range 0 5 y 5
ycas reported by Tannenbaum, the right-hand side of (11)is less
than one and we have C[ > C,. C[ only depends onC, and on how 6
varies with C at high concentrations.Therefore, for a given binary
system and a given C,, thevalue of C[ is fixed and can be
determined experimentally.
This idea may be clarified by the illustration of Fig. 1.Before
Tannenbaum, it was commonly assumed that theimpurity profile obeyed
the complementary error function(erfc) as shown by Curve I.
Tannenbaum showed that ifC,, the actual surface concentration, is
high, the actualprofile s something like Curve I1 instead of Curve
I.We are here pointing ou t that the tail of Curve I1 is
acomplementary error function, following ( lo) , in whichthe
Lapparent surface concentration C; is much higherthan theactual
concentration C,. Of course, (10) extendedabove the critical
concentration C, (Curve 111) is notvalid as a representation of the
actual profile of Curve 11.However, for computations which only
involve the tailof the curve where C < C,, (10) is a good
approximation.We have seen th at he ail of the actual
mpurityprofile can be represented by a complementary errorfunction
with an apparent surface concentration dif-ferent from the actual
value. It is interesting to note thatthe apparent, surface of the
complementary error
DISTANCE FROM SURFACE, X. IN MICRONSFig.2-Impurity profiles of
boron a nd phosphorus diffusions nsilicon a t 1200C.
function is precisely the actual surface. This nontrivialresult
enablesone t o calculate the low concentrationdiffusion oefficient
D L rom two experimental pointseven when concentration-dependent
diffusion is involved.Mathematically, it is straightforward t o
derive (10)from (1)under our conditions. Physically, it is not
obviousthat (10) should represent the tail of the impurity
profile.A solution like C = C, erfc (y - yc) (for C 5 C,) mightseem
t o be more readily acceptable than (10). However,this solution
isobviously incorrect because the plane,y = yc, is not a source of
constant concentration 6,.
EXPERIMENTSFig. 2 shows the experimental results of boron
andphosphorus diffusions in silicon at 1200C. It can be seen
that he experimental points fit (10) very well. Theimpurity
concentrations at different points were deter-mined by
measuringdiffused junction depths in siliconslices originally doped
with opposite types of impuritiesto various concentration levels.
By using (10) and choos-ing two points on each experimental curve,
we calculatedthe values of C[ under our diffusion conditions and
thevalues of D,. These calculated values are given in Fig. 2.D l
values obtained in this work are, for both phos-phorus and boron,
about one half of the values given byFuller and Dit~enberger,~ut n
good agreement with
7 C. S. Fuller and J. A. Ditzenberger. Diffusion of donor
andacceptor elements in silicon, J, p p l y P h y s . , vol. 27,
pp. 544-553;May, 1956.
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1965 W a l lma rk , et al.: Tunneles i s tor 359the boron data
obtained byKurtz and Yee and thephosphorus data given in a review
article by Smits.In he work of Fuller and Dit~enberger,~ he
urfaceconcentrations were of the order of 10 cm-3 and theeffect of
high impurity concentration on diffusionwasnot accounted forwhile
Kurtz and Yee obtained their
J . A p p l . Phys., vol. 31, pp. 303-305; February, 1960.Exakt
. natum., vol. 31, pp. 167-219; 1959.
8 A. D. Kurta and R. Yee,Diffusion of boron into silicon,F. M.
Smits, Diffusion in homoeopolaren Halbleitern, Ergeb.
boron data at low surface concentrations of the order of5 X IOC
M - ~ .
ACKNOWLEDGMENTThe author wishes t o express his gratitude t o
Dr. J. H.Forster for his interest and encouragement, and toDrs. J.
M. Early, R. M. Ryder and J. M. Goldey fortheir helpful comments on
the manuscript. He also wishes
t o express his appreciation to D. E. Iglesias for his
assis-tance n conducting the experiments, and to those col-leagues
who cooperated in various ways in carrying outthis work.
The Tunnel Resistor*Summary-Nonlinear load resistors in
high-speed tunnel-diodecomputer circuits offer several advantages
over conventionallinear load resistors, namely reduced power
dissipation and there-fore higher packing density, increased
switching speed and relaxedtolerances on the power supplies. Such
resistors have been con-structed by combining a tunnel-diode
junction with tunnelingleakage paths on the surface of the same
semiconductor junctionusing a metal plating technique. The plated
metal, which on theaverage is less than monoatomically thin, forms
conducting islandsthrough which tunneling takes place in parallel
with the tunnelingacross the junction. The added conductance is in
itself nonlinear.
The resulting characteristic exhibits a plateau where the
current issubstantially independent of voltage over a range
of50-1OOmv.The parallel resistance applied in this manner is free
of the spu-rious reactances usually connected with resistances
applied out-side or on the surface of the encapsulation of the
tunnel diode, andtherefore allows stable operation up to very high
frequencies.At the same t i e he application method allows the
necessary very
particularly useful as a load resistor for high-speed
digitaltunnel-diode circuits. It consists of a tunneling
junctionwith a thin layer of metal plated directly across it.The
plating method may also be used to alter thecharacteristics of
tunnel diodes, particularly t o raisethe peak current. The plating
is therefore, t o some extent,the inverse of etching, and may be
used in conjunctionwith etching to doctor the peak current t o
narrow toler-ances, but at a sacrifice in valley current.Device
Design
The current-voltage characteristic of a tunnel resistoris
shownby the solid linein Fig. 1.This type of character-istic is
often found in intermediate stages of fabricationof tunnel diodes
while they are being etched down t o the
INTRODUCTIONHILE LINEAR passivecomponents-resistors,capacitors
and inductors-are desirable for theoverwhelming number of
applications, special
advantages can sometimesbe obtained with nonlinearcomponents.1
This paper describes a simple, nonlineardevice, the tunnel
resistor, which combines the character-istics of a unnel diode and
a resistor. This device s1963.* Received May 17, 1963; revised
manuscript received July 8,t RCA Laboratories, Princeton, N. J.$
Semiconductor and Materials Div., RCA,Somerville, N. J.1 1 The
Technion, Haifa, Israel. Formerly with Electronic DataProcessing
Div., RCA, Pennsauken, N. J.sented a t Natl Electronics Conf.,
Chicago, Ill.; October 9-11, 1961.1 H. C. Lin, Nonlinear Resistance
for Microelectronics, pre-
the slope of the plateau could not be controlled simultane-ously
in fabrication.A better method t o obtain he desired
characteristic
is to combine in parallel a tunnel diode and a resistor,the
characteristics of which are shown dashed in Fig. 1.(In principle
this is what a semifinished tunnel dioderepresents.) Then the
resulting characteristic is obtainedthrough summation of currents
in Fig. 1. In this mannereachomponentmayeontrolled
independently,i.e., the tunnel diode may be etched to the
appropriatepeak current and the resistor plated to the
appropriateresistance value, t o arrive at a predetermined
plateaucurrent value, and simultaneously, a predetermined slopeof
the plateau.However, although the aver-all current-voltage
char-
